Geodesic
Control of the shape of a crack in an elastic body under the condition of possible contact of edges
Sibirskij žurnal industrialʹnoj matematiki, Tome 9 (2006) no. 2, pp. 20-30 
E. V. Vtorushin. Control of the shape of a crack in an elastic body under the condition of possible contact of edges. Sibirskij žurnal industrialʹnoj matematiki, Tome 9 (2006) no. 2, pp. 20-30. http://geodesic.mathdoc.fr/item/SJIM_2006_9_2_a2/
@article{SJIM_2006_9_2_a2,
     author = {E. V. Vtorushin},
     title = {Control of the shape of a~crack in an elastic body under the condition of possible contact of edges},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {20--30},
     year = {2006},
     volume = {9},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2006_9_2_a2/}
}
TY  - JOUR
AU  - E. V. Vtorushin
TI  - Control of the shape of a crack in an elastic body under the condition of possible contact of edges
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2006
SP  - 20
EP  - 30
VL  - 9
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SJIM_2006_9_2_a2/
LA  - ru
ID  - SJIM_2006_9_2_a2
ER  - 
%0 Journal Article
%A E. V. Vtorushin
%T Control of the shape of a crack in an elastic body under the condition of possible contact of edges
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2006
%P 20-30
%V 9
%N 2
%U http://geodesic.mathdoc.fr/item/SJIM_2006_9_2_a2/
%G ru
%F SJIM_2006_9_2_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

Under consideration is a homogeneous three-dimensional body with a crack in the form of a smooth surface. We impose some inequality constraints on the crack edges that describe their mutual nonpenetration. According to the Griffith criterion, the crack begins to propagate when the derivative of the energy functional with respect to the virtual increment of the crack surface area reaches a certain critical value. The value of this derivative depends, in particular, on the crack shape. The crack shape is determined that minimizes the value of the derivative of the energy functional; more precisely, the existence of a solution to the corresponding optimal control problem is proved.

[1] Khludnev A. M., Kovtunenko V. A., Analysis of Cracks in Solids, WIT-Press, Southampton, Boston, 2000

[2] Sokolowski J., Zolesio J. P., Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin, 1992 | MR

[3] Bochniak M. M., Sandig A.-M., Sensitivity analysis of stress intensity and notch factor in elastic structures, Preprint Stuttgärt Universitat; 97/54, SBF 404, Stuttgärt, 1997

[4] Mazya V. G., Nazarov S. A., “Asimptotika integralov energii pri malykh vozmuscheniyakh vblizi uglovykh i konicheskikh tochek”, Tr. Mosk. mat. o-va, 50, 1987, 79–129 | MR

[5] Othsuka K., “Mathematics of brittle fracture”, Theoretical Studies on Fracture Mechanics in Japan, Hiroshima-Denki Institute of Technology, Hiroshima, 1997, 99–172

[6] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno-gladkoi granitsei, Nauka, M., 1991

[7] Morozov N. F., Matematicheskie voprosy teorii treschin, Nauka, M., 1984 | MR

[8] Khludnev A. M., Ohtsuka K., Sokolowski J., “On derivative of energy functional for elastic bodies with cracks and unilateral conditions”, Quart. Appl. Math., 60 (2002), 99–109 | MR | Zbl

[9] Khludnev A. M., Sokolowski J., “On derivative of the energy functional in the elasticity theory”, Appl. Math. Mech., 64:3 (2000), 464–475 | MR

[10] Banichuk N. V., Optimizatsiya form uprugikh tel, Nauka, M., 1980 | MR

[11] Khludnev A. M., Sokolowski J., Modelling and control in solidmechanics, Birkhäuser, Basel e. a., 1997 | Zbl

[12] Botkin N., Khludnev A. M., Shape optimization for elastic structures with nonlinear cracks, Preprint Rheinische Friedrich-Wilhelms-Universität; 121, SFB 611, Bonn, 2003

[13] Homberg D., Khludnev A. M., “On safe crack shapes in elastic bodies”, European J. Mech. A Solids, 21 (2002), 991–998 | DOI | MR

[14] Khludnev A., Leontiev A., Herskovits J., “Nonsmooth domain optimization for elliptic equations with unilateral conditions”, J. Math. Pures Appl., 82 (2003), 197–212 | MR | Zbl

[15] Khludnev A. M., Sokolovski J., “Griffith formulae for elasticity systems with unilateral conditions in domains with cracks”, European J. Mech. A Solids, 19 (2000), 105–119 | DOI | MR | Zbl